Gravitational and electromagnetic radiation from binary black holes with electric and magnetic charges: Elliptical orbits on a cone

Magnetic charges, if they exist in the early Universe, will provide a hitherto unexplored window to probe fundamental physics in the Standard Model of particle physics and beyond. Though no evidence of magnetic charges has been found yet  Staelens (2019); Mavromatos and Mitsou (2020), magnetically charged black holes have attracted much attention not only in theoretical study but also in recent astronomical observations Maldacena (2020); Bai et al. (2020); Ghosh et al. (2020). Recently, spectacular properties of magnetic charged black holes have been extensively discussed in Maldacena (2020). It is shown there that the magnetic field near the horizon of a magnetic black hole could be strong enough to restore the electroweak symmetry. The phenomenology of low-mass magnetic black holes, which can have electroweak-symmetric coronas outside of the event horizon, has been comprehensively studied in Bai et al. (2020). Potential astrophysical signatures for magnetically charged black holes have also been investigated in Ghosh et al. (2020).

According to the “no-hair” conjecture, a general relativistic black hole is completely described by four parameters: mass, angular momentum, magnetic charge as well as electric charge. Compared to Schwarzschild black holes, charged black holes have rich phenomena. Recently, charged black holes have been discussed extensively Cardoso et al. (2016); Liebling and Palenzuela (2016); Toshmatov et al. (2018); Bai and Orlofsky (2020); Allahyari et al. (2020); Liu et al. (2020a); Christiansen et al. (2020); Wang et al. (2020); Bozzola and Paschalidis (2020); Kim and Kobakhidze (2020); Cardoso et al. (2020a); Christiansen (2020); Cardoso et al. (2020b); McInnes (2020). Binary black holes with charges emit not only gravitational radiation but also electromagnetic radiation. By using a Newtonian method with the inclusion of radiation reaction, a previous study Liu et al. (2020a) obtains the evolutions of orbits and calculates merger times of binary black holes with electric charges by considering the Keplerian motion of two charged bodies and accounting for the loss of energy and angular momentum due to the emission of gravitational and electromagnetic waves. The Coulomb-type force due to a pure electric or magnetic charge changes the coupling parameter of the gravitational force and alters the gravitational wave emission compared to an uncharged binary, which leads to a different merger rate of primordial black hole Liu et al. (2020a). The bias in the binary parameters due to the charge-chirp mass degeneracy is discussed in Christiansen et al. (2020). For the first merger event of binary black holes reported by LIGO/Virgo, GW150914, Ref. Liebling and Palenzuela (2016) argues that the magnetic charge should be small, and it is shown in Ref. Wang et al. (2020) that binary black holes could have some electric charge by using a full Bayesian analysis with Gaussian noise.

A dyonic black hole is a nonrotating or rotating black hole with an electric charge $q$ and a magnetic charge $g$. A dyonic nonrotating black hole has the same metric as the Reissner-Nordström black hole with $q^{2}$ replaced by $q^{2}+g^{2}$ Kasuya (1982). In the Minkowski spacetime, the nonrelativistic interaction of two dyons was studied in a quantum theory Zwanziger (1968) and in classical theory Schwinger et al. (1976) . In the previous paper Liu et al. (2020b), which will be denoted as I, we have derived the equations of motion of dyonic black hole binaries and explored features of static orbits (without radiation). In Ref. I, static orbits of dyonic black hole binaries on a conic section are divided into three categories: (1) $e=0$; (2) $e\neq 0$ and $\sin\theta$ is rational; (3) $e\neq 0$ and $\sin\theta$ is irrational ($\theta$ is the conic angle, $e$ is the eccentricity with the definition in Eq. (10).), and the orbits of dyonic black hole binaries in those different cases have different topology. In the first case of $e=0$, the three-dimensional trajectory is a two-dimensional circular orbit. In the second case when $e\neq 0$ and $\sin\theta$ is rational, the orbit is closed and confined to the surface of a cone. In the last case when $e\neq 0$ and $\sin\theta$ is irrational, the conic-shaped orbit of the binary is not closed and shows a chaotic behavior of a conserved autonomous system. For circular orbits, in Ref. I we have calculated the total emission rate of energy and angular momentum due to gravitational and electromagnetic radiation. Furthermore, the merger times of dyonic binaries for circular orbits are calculated.

In the universe, most binary black hole systems have non-zero eccentricity and can have even large eccentricity for those from encounters of black holes. Indeed the circular orbits are a very rare and special case. Therefore, it is important and meaningful to explore the evolutions of elliptical orbits for binary black holes with electric and magnetic charges and their characteristic features. In this paper, we explore the evolutions of elliptical orbits for binary black holes with electric and magnetic charges by considering the equation of motion of two dyonic bodies and accounting for a loss of energy via quadrupolar emission of gravitational waves and dipolar emission of electromagnetic ones. For dyonic binaries, in the 0th order post-Newtonian (PN) approximation, the angular-momentum-dependent and non-central Lorentz force cause the orbits to execute three-dimensional and complex trajectories. We find that the emission rates of energy and angular momentum have the same dependence of $\theta$ for all different cases. Moreover, we find a circular orbit remains circular and an elliptic orbit becomes quasi-circular because of electromagnetic and gravitational radiation. Using the evolution of orbits, we derive the waveform models for dyonic binary black hole inspirals.

The organization of this paper as follows. In Sec. II, we work out the total emission rate of angular momentum and energy due to gravitational and electromagnetic radiation. In Sec. III, we derive the evolutions of orbits and find a circular orbit remains circular and an elliptic orbit becomes quasi-circular. In Sec. IV, we obtain the waveforms for dyonic binary black hole inspirals. In Sec. V, we discuss the physical implications and conclude the perspective of the dyonic black hole binaries. Throughout this work, we set $G=c=4\pi\varepsilon_{0}=\frac{\mu_{0}}{4\pi}=1$.

In this paper, we adopt the the weak-field approximation. In other words, we only focus on the case that the distance of the dyonic black hole binary is much larger than their event horizons. In such a case, a dyonic black hole binary during the inspiral motion whose distance is much larger than their event horizons is well approximated by a pair of massive point-like objects with electric and magnetic charges.

The elliptical orbits on a cone which precess around the generalized angular momentum are a characteristic feature of the dyonic binary with both electric and magnetic charges. We extend the previous works Liu et al. (2020b) to inspiral elliptical orbits on a cone, find the gravitational radiation á la Peters and Mathews (1963); Peters (1964) and the synchrotron radiation due to electric and magnetic dipoles á la Landau and Lifshitz (1975) and investigate the radiation reaction on the orbital motion.

Now that we have the emissions of energy and angular momentum due to gravitational radiation and electromagnetic radiation, we can be able to calculate the evolution of the orbit through two Keplerian parameters $a$, $e$ and another parameter $\theta$ due to the presence of a magnetic charge.

Though we have the emissions of energy  (40) and angular momentum (41), we have three parameters $a,e,$ and $\theta$. According to Subsection II.1 and II.2, when we consider gravitational and electromagnetic radiation, we find only $\dot{L}^{3}\neq 0$, which implies that the direction of $\bm{L}$ does not change while the magnitude of $\bm{L}$ decreases. From Eq. (10) and $\tan(\theta)=\tilde{L}/|D|$, we have the relation of $a$, $e$ and $\theta$:

$\displaystyle\tan^{2}(\theta)=\frac{\mu(-C)a(1-e^{2})}{D^{2}}.$

(42)

Using the relation of $a$, $e$ and $\theta$, we may rewrite Eqs. (40) and (41) as functions of $a$ and $e$ as

	$\displaystyle\left\langle\frac{dE}{dt}\right\rangle$	$\displaystyle=$	$\displaystyle\frac{((\Delta\sigma_{q})^{2}+(\Delta\sigma_{g})^{2})(-C)\left(4a\left(e^{4}+e^{2}-2\right)(-C)\mu-D^{2}\left(3e^{4}+24e^{2}+8\right)\right)}{12a^{5}\left(1-e^{2}\right)^{7/2}\mu}$		(43)
	$\displaystyle-\frac{(-C)^{5/2}\left(8a^{2}h_{1}C^{2}\mu^{2}-3aD^{2}h_{2}(-C)\mu+3D^{4}h_{3}\right)}{120a^{11/2}\left(1-e^{2}\right)^{4}\mu^{3/2}\left(D^{2}+a\left(1-e^{2}\right)(-C)\mu\right)^{3/2}},$

	$\displaystyle\left\langle\frac{dL}{dt}\right\rangle$	$\displaystyle=$	$\displaystyle-\frac{((\Delta\sigma_{q})^{2}+(\Delta\sigma_{g})^{2})(-C)\left(D^{2}\left(e^{2}+2\right)+2a\left(1-e^{2}\right)(-C)\mu\right)}{3a^{3}\left(1-e^{2}\right)^{3/2}\sqrt{\left(D^{2}+a\left(1-e^{2}\right)(-C)\mu\right)}}$		(44)
	$\displaystyle-\frac{(-C)\left(16a^{2}h_{4}C^{2}\mu^{2}-aD^{2}h_{5}(-C)\mu+D^{4}h_{6}\right)}{20a^{5}\left(1-e^{2}\right)^{7/2}\mu^{2}\sqrt{\left(a\left(1-e^{2}\right)(-C)\mu+D^{2}\right)}},$

where the first term is the energy (angular momentum) loss rate due to electromagnetic radiation and the second term is the energy (angular momentum) loss rate due to gravitational radiation. Here, we have used the short-hand notations

	$\displaystyle h_{1}$	$\displaystyle=$	$\displaystyle\left(1-e^{2}\right)^{2}(37e^{4}+292e^{2}+96),$
	$\displaystyle h_{2}$	$\displaystyle=$	$\displaystyle 45e^{8}+1005e^{6}+670e^{4}-1448e^{2}-272,$
	$\displaystyle h_{3}$	$\displaystyle=$	$\displaystyle 5e^{6}+90e^{4}+120e^{2}+16$		(45)

and

	$\displaystyle h_{4}$	$\displaystyle=$	$\displaystyle\left(1-e^{2}\right)^{2}(7e^{2}+8),$
	$\displaystyle h_{5}$	$\displaystyle=$	$\displaystyle 31e^{6}+297e^{4}-192e^{2}-136,$
	$\displaystyle h_{6}$	$\displaystyle=$	$\displaystyle 3(e^{2}+8)e^{2}+8.$		(46)

In principle, we can also rewrite Eqs. (40) and (41) as functions of $a$ and $\theta$ or functions of $e$ and $\theta$. When we rewrite Eqs. (40) and (41) as functions of Keplerian parameters $a$ and $e$, it is much more easily to come back to the results of Liu et al. (2020a); Peters (1964). To compare the differences of the energy (angular momentum) loss rate between circular orbits and elliptical orbits, we define

$\displaystyle f_{1}(e)=\left\langle\frac{dE_{GW}}{dt}\right\rangle/\left\langle\frac{dE_{GW}}{dt}\right\rangle|_{e=0},$

(47)

$\displaystyle f_{2}(e)=\left\langle\frac{dE_{EM}}{dt}\right\rangle/\left\langle\frac{dE_{EM}}{dt}\right\rangle|_{e=0},$

(48)

and

$\displaystyle f_{3}(e)=\left\langle\frac{dL_{GW}}{dt}\right\rangle/\left\langle\frac{dL_{GW}}{dt}\right\rangle|_{e=0},$

(49)

$\displaystyle f_{4}(e)=\left\langle\frac{dL_{EM}}{dt}\right\rangle/\left\langle\frac{dL_{EM}}{dt}\right\rangle|_{e=0}.$

(50)

In Fig. 1, we plot $f_{1}(e)$, $f_{2}(e)$, $f_{3}(e)$ and $f_{4}(e)$ as functions of $e$ by choosing $m_{1}=m_{2}=m$, $q_{1}=q_{2}=0.2m$, $g_{1}=-g_{2}=0.1m$ and $a=10^{4}m$. From Fig. 1, we find that the energy and angular momentum loss rates increase quite fast as the eccentricity increases. Thus, highly elliptical orbits lose the energy and angular momentum more rapidly than the less elliptical orbits.

Now, we find the evolution of $a$ and $e$. From the chain rule for differentiation, we have

$\displaystyle\frac{dE}{da}\frac{da}{dt}=\left\langle\frac{dE}{dt}\right\rangle,$

(51)

$\displaystyle\frac{dL}{da}\frac{da}{dt}+\frac{dL}{de}\frac{de}{dt}=\left\langle\frac{dL}{dt}\right\rangle.$

(52)

For simplicity, we divide the rates of the semimajor axis and eccentricity into two parts: the first part is the loss rates due to electromagnetic radiation and the second part is the loss rates due to gravitational radiation. In other words,

$\displaystyle\frac{da}{dt}=\frac{da_{EM}}{dt}+\frac{da_{GW}}{dt},$

(53)

$\displaystyle\frac{de}{dt}=\frac{de_{EM}}{dt}+\frac{de_{GW}}{dt},$

(54)

where $\frac{da_{EM}}{dt},\frac{da_{GW}}{dt},\frac{de_{EM}}{dt}$ and $\frac{de_{GW}}{dt}$ satisfy, respectively,

$\displaystyle\frac{dE}{da}\frac{da_{EM}}{dt}=\left\langle\frac{dE_{EM}}{dt}\right\rangle,$

(55)

$\displaystyle\frac{dL}{da}\frac{da_{EM}}{dt}+\frac{dL}{de}\frac{de_{EM}}{dt}=\left\langle\frac{dL_{EM}}{dt}\right\rangle,$

(56)

and

$\displaystyle\frac{dE}{da}\frac{da_{GW}}{dt}=\left\langle\frac{dE_{GW}}{dt}\right\rangle,$

(57)

$\displaystyle\frac{dL}{da}\frac{da_{GW}}{dt}+\frac{dL}{de}\frac{de_{GW}}{dt}=\left\langle\frac{dL_{GW}}{dt}\right\rangle,$

(58)

Using

$\displaystyle E=\frac{C}{2a},$

(59)

and

$\displaystyle L=\sqrt{{a\left(1-e^{2}\right)(-C)\mu}+D^{2}},$

(60)

we finally obtain

$\displaystyle\frac{da_{GW}}{dt}=\frac{(-C)^{3/2}\left(-8a^{2}\left(e^{2}-1\right)^{2}h_{1}C^{2}\mu^{2}+3aD^{2}h_{2}(-C)\mu-3D^{4}h_{3}\right)}{60a^{7/2}\left(e^{2}-1\right)^{4}\left(\mu\left(D^{2}-a\left(e^{2}-1\right)(-C)\mu\right)\right)^{3/2}},$

(61)

$\displaystyle\frac{de_{GW}}{dt}=-\frac{e\Bigl{(}8a^{2}h_{7}C^{2}\mu^{2}-3aD^{2}h_{8}(-C)\mu+33D^{4}h_{9}\Bigr{)}}{120a^{6}\left(1-e^{2}\right)^{9/2}\mu^{3}},$

(62)

$\displaystyle\frac{da_{EM}}{dt}=-\frac{(\Delta\sigma_{q})^{2}+(\Delta\sigma_{g})^{2})\Bigl{(}D^{2}\left(3e^{4}+24e^{2}+8\right)-4a\left(e^{4}+e^{2}-2\right)(-C)\mu\Bigr{)}}{6a^{3}\left(1-e^{2}\right)^{7/2}\mu},$

(63)

$\displaystyle\frac{de_{EM}}{dt}=-\frac{((\Delta\sigma_{q})^{2}+(\Delta\sigma_{g})^{2})e\left(7D^{2}\left(e^{2}+4\right)+12a\left(1-e^{2}\right)(-C)\mu\right)}{12a^{4}\left(1-e^{2}\right)^{5/2}\mu},$

(64)

where

	$\displaystyle h_{7}$	$\displaystyle=$	$\displaystyle\left(1-e^{2}\right)^{2}\left(121e^{2}+304\right),$
	$\displaystyle h_{8}$	$\displaystyle=$	$\displaystyle 107e^{6}+1537e^{4}-308e^{2}-1336,$
	$\displaystyle h_{9}$	$\displaystyle=$	$\displaystyle e^{4}+12e^{2}+8.$		(65)

Using Eqs. (53), (54), (61), (62), (63) and (64), we can obtain $\frac{da}{dt}$ and $\frac{de}{dt}$ as functions of $a$ and $e$ which means we find the evolutions of orbits. Notice that $0\leq e<1$ and $C<0$, for arbitrary $q_{1},q_{2},g_{1},g_{2},m_{1}$ and $m_{2}$, we always have $\frac{da_{GW}}{dt}<0$, $\frac{de_{GW}}{dt}\leq 0$, $\frac{da_{EM}}{dt}<0$ and $\frac{de_{EM}}{dt}\leq 0$ which imply $\frac{da}{dt}<0$ and $\frac{de}{dt}\leq 0$. If $e=0$, we have

$\displaystyle\frac{da_{GW}}{dt}=-\frac{4\left(a(-C)\mu+D^{2}\right)\left(16a(-C)\mu+D^{2}\right)}{5a^{5}\mu^{3}},$

(66)

$\displaystyle\frac{da_{EM}}{dt}=-\frac{4((\Delta\sigma_{q})^{2}+(\Delta\sigma_{g})^{2})\left(a(-C)\mu+D^{2}\right)}{3a^{3}\mu},$

(67)

$\displaystyle\frac{de_{GW}}{dt}=\frac{de_{EM}}{dt}=0,$

(68)

which are consistent with Liu et al. (2020b). Therefore a circular orbit remains circular. For $e>0$, we have $\frac{de}{dt}=\frac{de_{GW}}{dt}+\frac{de_{EM}}{dt}<0$ instead, and therefore an elliptical orbit becomes quasi-circular because of electromagnetic and gravitational radiations. Here, we notice that

$\displaystyle\frac{da_{GW}}{dt}/\frac{da_{GW}}{dt}|_{e=0}=\left\langle\frac{dE_{GW}}{dt}\right\rangle/\left\langle\frac{dE_{GW}}{dt}\right\rangle|_{e=0}=f_{1}(e),$

$\displaystyle\frac{da_{EM}}{dt}/\frac{da_{EM}}{dt}|_{e=0}=\left\langle\frac{dL_{GW}}{dt}\right\rangle/\left\langle\frac{dL_{GW}}{dt}\right\rangle|_{e=0}=f_{3}(e).$

In this subsection, we obtain the evolutions of orbits in three-dimensional trajectories. Next, we will show our results are also valid for orbits confined in $x$-$y$ plane, in other words, $D=0$, which corresponds to purely electric or magnetic charges or $q_{2}/q_{1}=g_{2}/g_{1}$ of balancing out the velocity-dependent Lorentz forces. When $D=0$, we find

$\displaystyle\frac{da_{GW}}{dt}=-\frac{2\left(37e^{4}+292e^{2}+96\right)C^{2}}{15a^{3}\left(1-e^{2}\right)^{7/2}\mu},$

(71)

$\displaystyle\frac{de_{GW}}{dt}=-\frac{e\left(121e^{2}+304\right)C^{2}}{15a^{4}\left(1-e^{2}\right)^{5/2}\mu},$

(72)

$\displaystyle\frac{da_{EM}}{dt}=-\frac{2((\Delta\sigma_{q})^{2}+(\Delta\sigma_{g})^{2})\left(e^{2}+2\right)(-C)}{3a^{2}\left(1-e^{2}\right)^{5/2}},$

(73)

$\displaystyle\frac{de_{EM}}{dt}=-\frac{((\Delta\sigma_{q})^{2}+(\Delta\sigma_{g})^{2})e(-C)}{a^{3}\left(1-e^{2}\right)^{3/2}},$

(74)

which are consistent with Liu et al. (2020a); Christiansen et al. (2020). In particular, for Schwarzschild black holes, $g_{1}=g_{2}=q_{1}=q_{2}=0$, which imply $\Delta\sigma_{q}=\Delta\sigma_{g}=D=0$ and $C=-\mu M$. Then we get

$\displaystyle\frac{da_{GW}}{dt}=-\frac{2\left(37e^{4}+292e^{2}+96\right)\mu M^{2}}{15a^{3}\left(1-e^{2}\right)^{7/2}},$

(75)

$\displaystyle\frac{de_{GW}}{dt}=-\frac{e\left(121e^{2}+304\right)\mu M^{2}}{15a^{4}\left(1-e^{2}\right)^{5/2}},$

(76)

$\displaystyle\frac{da_{EM}}{dt}=\frac{de_{EM}}{dt}=0,$

(77)

which are consistent with Peters and Mathews (1963); Peters (1964). According to Section III and Appendix, we can conclude a few features of the evolution of orbits.

1. 1.

   For arbitrary cases, the semimajor axis $a$ always decreases with time due to the energy loss in gravitational and electromagnetic waves.

2. 2.

   A circular orbit remains circular while an elliptical orbit becomes quasi-circular due to the loss of energy and angular momentum. In other words, the effect of the back-reaction of gravitational and electromagnetic waves is to circularize the orbit.

3. 3.

   When $D=0$, the conic angle $\theta$ keeps the value $\theta=\pi/2$ which means the orbit is confined in $x$-$y$ plane. When $D\neq 0$, with the semimajor axis $a$ shrinking, the conic angle $\theta$ decreases. When the semimajor axis $a$ shrinks to nearly zero, the conic angle $\theta$ also decreases to nearly zero.